### Nuprl Lemma : rinv-functionality-lemma

`∀x,y:ℤ. ∀a,b,n:ℕ+.`
`  ((n ≤ (a * |x|)) `` (n ≤ (b * |y|)) `` (|x - y| ≤ 4) `` (|((4 * n * n) ÷ x) - (4 * n * n) ÷ y| ≤ (2 + (16 * a * b))))`

Proof

Definitions occuring in Statement :  absval: `|i|` nat_plus: `ℕ+` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` divide: `n ÷ m` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` absval: `|i|` nat_plus: `ℕ+` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` int_nzero: `ℤ-o` subtract: `n - m` le: `A ≤ B` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` cand: `A c∧ B` uiff: `uiff(P;Q)` less_than': `less_than'(a;b)`
Lemmas referenced :  mul_com absval_sym rem_bounds_absval_le mul_preserves_le absval_pos nat_plus_subtype_nat mul_bounds_1a multiply_functionality_wrt_le absval-diff-symmetry false_wf int_term_value_add_lemma itermAdd_wf multiply-is-int-iff add_functionality_wrt_eq int-triangle-inequality add_functionality_wrt_le le_weakening le_functionality decidable__le add-commutes add-swap one-mul mul-associates mul-commutes mul-swap minus-one-mul add-associates minus-add mul-distributes div_rem_sum2 nequal_wf int_term_value_subtract_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma itermSubtract_wf intformeq_wf intformnot_wf decidable__equal_int iff_weakening_equal absval_mul true_wf squash_wf nat_plus_wf nat_wf le_wf int_entire_a absval_nat_plus subtract_wf absval_wf mul_cancel_in_le equal_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_and_lemma intformless_wf itermConstant_wf itermMultiply_wf itermVar_wf intformle_wf intformand_wf satisfiable-full-omega-tt nat_plus_properties int_subtype_base subtype_base_sq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin instantiate lemma_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination hypothesisEquality equalityTransitivity equalitySymmetry independent_functionElimination sqequalRule setElimination rename natural_numberEquality minusEquality dependent_pairFormation lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll because_Cache divideEquality multiplyEquality applyEquality addEquality imageElimination imageMemberEquality baseClosed universeEquality productElimination unionElimination dependent_set_memberEquality remainderEquality pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}x,y:\mBbbZ{}.  \mforall{}a,b,n:\mBbbN{}\msupplus{}.
((n  \mleq{}  (a  *  |x|))
{}\mRightarrow{}  (n  \mleq{}  (b  *  |y|))
{}\mRightarrow{}  (|x  -  y|  \mleq{}  4)
{}\mRightarrow{}  (|((4  *  n  *  n)  \mdiv{}  x)  -  (4  *  n  *  n)  \mdiv{}  y|  \mleq{}  (2  +  (16  *  a  *  b))))

Date html generated: 2016_05_18-AM-06_54_19
Last ObjectModification: 2016_01_17-AM-01_47_50

Theory : reals

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